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Properties

Label 2-990-11.4-c1-0-16
Degree $2$
Conductor $990$
Sign $-0.946 + 0.323i$
Analytic cond. $7.90518$
Root an. cond. $2.81161$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.809 − 0.587i)2-s + (0.309 + 0.951i)4-s + (0.809 − 0.587i)5-s + (−0.133 − 0.410i)7-s + (0.309 − 0.951i)8-s − 10-s + (−1.92 − 2.69i)11-s + (−3.19 − 2.31i)13-s + (−0.133 + 0.410i)14-s + (−0.809 + 0.587i)16-s + (2.76 − 2.01i)17-s + (−1.89 + 5.82i)19-s + (0.809 + 0.587i)20-s + (−0.0276 + 3.31i)22-s − 4.17·23-s + ⋯
L(s)  = 1  + (−0.572 − 0.415i)2-s + (0.154 + 0.475i)4-s + (0.361 − 0.262i)5-s + (−0.0504 − 0.155i)7-s + (0.109 − 0.336i)8-s − 0.316·10-s + (−0.581 − 0.813i)11-s + (−0.884 − 0.642i)13-s + (−0.0356 + 0.109i)14-s + (−0.202 + 0.146i)16-s + (0.671 − 0.487i)17-s + (−0.434 + 1.33i)19-s + (0.180 + 0.131i)20-s + (−0.00588 + 0.707i)22-s − 0.869·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.946 + 0.323i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.946 + 0.323i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(990\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $-0.946 + 0.323i$
Analytic conductor: \(7.90518\)
Root analytic conductor: \(2.81161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{990} (631, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 990,\ (\ :1/2),\ -0.946 + 0.323i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0979479 - 0.588944i\)
\(L(\frac12)\) \(\approx\) \(0.0979479 - 0.588944i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 + 0.587i)T \)
3 \( 1 \)
5 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (1.92 + 2.69i)T \)
good7 \( 1 + (0.133 + 0.410i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (3.19 + 2.31i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-2.76 + 2.01i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.89 - 5.82i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 4.17T + 23T^{2} \)
29 \( 1 + (-0.195 - 0.602i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (8.34 + 6.06i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.26 + 6.98i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.821 - 2.52i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 11.5T + 43T^{2} \)
47 \( 1 + (0.917 - 2.82i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (4.24 + 3.08i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (3.35 + 10.3i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (8.51 - 6.18i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 10.9T + 67T^{2} \)
71 \( 1 + (8.29 - 6.03i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (3.44 + 10.5i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-6.02 - 4.37i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-6.37 + 4.63i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 1.34T + 89T^{2} \)
97 \( 1 + (-1.41 - 1.03i)T + (29.9 + 92.2i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.680842604966913295426897446316, −8.929937075727123677697826447306, −7.83570955817378833841182927614, −7.55520229432354982140425297675, −6.02806577501072289612688179325, −5.43690681386681491164663160779, −4.07704171925978104786440846064, −3.01346777552907193160885261172, −1.89236836812356597118340539089, −0.31100919850633050087641862713, 1.77598355005233069722962668646, 2.77574711441297523762297559755, 4.38668097711083593148956850035, 5.28208037253866302664777996992, 6.23652624503821787352393305525, 7.17042355730318431027249777301, 7.66021014836705419093436731570, 8.877161080075566300941180185626, 9.408590670007452837928236971419, 10.32185646806555138017258853292

Graph of the $Z$-function along the critical line